The Resultant of Developed Systems of Laurent Polynomials

Let $R_{\mathrm{\Delta }}(f_1,\dots ,f_{n+1})$ be the {\it $\mathrm{\Delta }$-resultant} (see below) of $(n+1)$-tuple of Laurent polynomials. We provide an algorithm for computing $R_{\mathrm{\Delta }}$ assuming that an $n$-tuple $(f_2,\dots ,f_{n+1})$ is {\it developed} (see sec.6). We provide a relation between the product of $f_1$ over roots of $f_2=\cdots =f_{n+1}=0$ in $({\mathbb{C}}^{\ast }{)}^n$ and the product of $f_2$ over roots of $f_1=f_3=\cdots =f_{n+1}=0$ in $({\mathbb{C}}^{\ast }{)}^n$ assuming that the $n$-tuple $(f_1{f}_2,f_3,\dots ,f_{n+1})$ is developed. If all $n$-tuples contained in $(f_1,\dots ,f_{n+1})$ are developed we provide a signed version of Poisson formula for $R_{\mathrm{\Delta }}$. In our proofs we use a topological arguments and topological version of the Parshin reciprocity laws.